We apply the improved (G′/G) expansion method to the Vakhnenko-Parkes equation. As a result, many new and more general exact solutions have been obtained for the equation. Comparing our solutions with those gained by other authors indicates that the improved (G′/G) expansion is more effective in solving the general solutions to differential equations.

Citric acid is the primary organic acid that affects the taste of strawberry fruit. Glycolysis supplies key substrates for the tricarboxylic acid cycle (TCA cycle). However, little is known about the regulatory mechanisms of glycolytic genes on citric acid metabolism in strawberry fruits. In this study, the citric acid content of strawberry fruit displayed a trend of rising and decreasing from the initial red stage to the full red stage and then dark red stage. Thus, a difference in citric acid metabolic regulation was suspected during strawberry fruit development. In addition, overexpression of either cytoplasm glyceraldehyde-3-phosphate dehydrogenase (FxaC_14g13400, namely FaGAPC2) or pyruvate kinase (FxaC_15g00080, namely FaPKc2.2) inhibited strawberry fruit ripening and the accumulation of citric acid, leading to a range of maturity stages from partial red to full red stage. The combined transcriptome and metabolome analysis revealed that overexpression of FaGAPC2 and FaPKc2.2 significantly suppressed the expression of phosphoenolpyruvate carboxykinase (FxaC_1g21491, namely FaPEPCK) but enhanced the content of glutamine and aspartic acid. Meanwhile, the activities of PEPCK and glutamate decarboxylase (GAD) were inhibited, but the activities of glutamine synthase (GS) were increased in FaGAPC2/FaPKc2.2-overexpressed fruit. Further, functional verification demonstrated that overexpression of FaPEPCK can promote strawberry fruit ripening, resulting in a range of maturity stage from full red to dark red stage, while the citric acid synthase (CS) activities and citric acid content were significantly decreased. Overall, this study revealed that FaGAPC2/FaPKc2.2 and FaPEPCK perform an important role in reducing citric acid content in strawberry fruit, and FaGAPC2/FaPKc2.2 mainly by promoting the GS degradation pathway and FaPEPCK mainly by inhibiting the CS synthesis pathway.

Ascorbic acid (AsA) is a crucial water-soluble antioxidant in strawberry fruit, but limited research is currently available on the identification and functional validation of key genes involved in AsA metabolism in strawberries. This study analyzed the FaMDHAR gene family identification, which includes 168 genes. Most of the products of these genes are predicted to exist in the chloroplast and cytoplasm. The promoter region is rich in cis-acting elements related to plant growth and development, stress and light response. Meanwhile, the key gene FaMDHAR50 that positively regulates AsA regeneration was identified through comparative transcriptome analysis of ‘Benihoppe’ strawberry (WT) and its natural mutant (MT) with high AsA content (83 mg/100 g FW). The transient overexpression experiment further showed that overexpression of FaMDHAR50 significantly enhanced the AsA content by 38% in strawberry fruit, with the upregulated expression of structural genes involved in AsA biosynthesis (FaGalUR and FaGalLDH) and recycling and degradation (FaAPX, FaAO and FaDHAR) compared with that of the control. Moreover, increased sugar (sucrose, glucose and fructose) contents and decreased firmness and citric acid contents were observed in the overexpressed fruit, which were accompanied by the upregulation of FaSNS, FaSPS, FaCEL1 and FaACL, as well as the downregulation of FaCS. Additionally, the content of pelargonidin 3-glucoside markedly decreased, while cyanidin chloride increased significantly. In summary, FaMDHAR50 is a key positive regulatory gene involved in AsA regeneration in strawberry fruit, which also plays an important role in the formation of fruit flavor, apperance and texture during strawberry fruit ripening.

By using the theory of planar dynamical systems to a coupled nonlinear wave equation, the existence of bell-shaped solitary wave solutions, kink-shaped solitary wave solutions, and periodic wave solutions is obtained. Under the different parametric values, various sufficient conditions to guarantee the existence of the above solutions are given. With the help of three different undetermined coefficient methods, we investigated the new exact explicit expression of all three bell-shaped solitary wave solutions and one kink solitary wave solutions with nonzero asymptotic value for a coupled nonlinear wave equation. The solutions cannot be deduced from the former references.